Non-uniform bound and finite time blow up for solutions to a drift–diffusion equation in higher dimensions

We show the non-uniform bound for a solution to the Cauchy problem of a drift–diffusion equation of a parabolic–elliptic type in higher space dimensions. If an initial data satisfies a certain condition involving the entropy functional, then the corresponding solution to the equation does not remain uniformly bounded in a scaling critical space. In other words, the solution grows up at [Formula: see text] in the critical space or blows up in a finite time. Our presenting results correspond to the finite time blowing up result for the two-dimensional case. The proof relies on the logarithmic entropy functional and a generalized version of the Shannon inequality. We also give the sharp constant of the Shannon inequality.